Coordinate systems/Derivation of formulas/Original Copy from Wikipedia

Table from Wikipedia

• from https://en.wikipedia.org/w/index.php?title=Del_in_cylindrical_and_spherical_coordinates&oldid=612099218
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Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates

Operation	Cartesian coordinates (x, y, z)	Cylindrical coordinates (ρ, ϕ, z)	Spherical coordinates (r, θ, ϕ)	Parabolic cylindrical coordinates (σ, τ, z)
Definition of coordinates	$\begin{array}{cc}\rho & =\sqrt{x^2+y^2}\\ \varphi & =\arctan (y/x)\\ z& =z\end{array}$	$\begin{array}{cc}x& =\rho \cos \varphi \\ y& =\rho \sin \varphi \\ z& =z\end{array}$	$\begin{array}{cc}x& =r\sin \theta \cos \varphi \\ y& =r\sin \theta \sin \varphi \\ z& =r\cos \theta \end{array}$	$\begin{array}{cc}x& =\sigma \tau \\ y& =\frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =z\end{array}$
	$\begin{array}{cc}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arccos (z/r)\\ \varphi & =\arctan (y/x)\end{array}$	$\begin{array}{cc}r& =\sqrt{{\rho }^2+z^2}\\ \theta & =\arctan (\rho /z)\\ \varphi & =\varphi \end{array}$	$\begin{array}{cc}\rho & =r\sin \theta \\ \varphi & =\varphi \\ z& =r\cos \theta \end{array}$	$\begin{array}{cc}\rho \cos \varphi & =\sigma \tau \\ \rho \sin \varphi & =\frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =z\end{array}$
Definition of unit vectors	$\begin{array}{cc}\hat{\mathit{\rho }}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \hat{\mathit{\varphi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\cos \varphi \hat{\mathit{\rho }}-\sin \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐲}& =\sin \varphi \hat{\mathit{\rho }}+\cos \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\sin \theta \cos \varphi \widehat{𝒓}+\cos \theta \cos \varphi \hat{\mathit{\theta }}-\sin \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐲}& =\sin \theta \sin \varphi \widehat{𝒓}+\cos \theta \sin \varphi \hat{\mathit{\theta }}+\cos \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝒓}-\sin \theta \hat{\mathit{\theta }}\end{array}$	$\begin{array}{cc}\hat{\mathit{\sigma }}& =\frac{\tau \widehat{𝐱}-\sigma \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \hat{\mathit{\tau }}& =\frac{\sigma \widehat{𝐱}+\tau \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$
	$\begin{array}{cc}\widehat{𝐫}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}+z\widehat{𝐳}}{\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{xz\widehat{𝐱}+yz\widehat{𝐲}-(x^2+y^2)\widehat{𝐳}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\varphi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\end{array}$	$\begin{array}{cc}\widehat{𝐫}& =\frac{\rho \hat{\mathit{\rho }}+z\widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{z\hat{\mathit{\rho }}-\rho \widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}\hat{\mathit{\rho }}& =\sin \theta \widehat{𝐫}+\cos \theta \hat{\mathit{\theta }}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝐫}-\sin \theta \hat{\mathit{\theta }}\end{array}$	$\begin{array}{}\end{array}$
A vector field $𝐀$	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\widehat{𝐳}$	$A_r\widehat{𝒓}+A_{\theta }\hat{\mathit{\theta }}+A_{\varphi }\hat{\mathit{\varphi }}$	$A_{\sigma }\hat{\mathit{\sigma }}+A_{\tau }\hat{\mathit{\tau }}+A_{\varphi }\widehat{𝐳}$
Gradient $\mathrm{\nabla }f$	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial r}\widehat{𝒓}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}$	$\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \sigma }\hat{\mathit{\sigma }}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \tau }\hat{\mathit{\tau }}+\frac{\partial f}{\partial z}\widehat{𝐳}$
Divergence $\mathrm{\nabla }\cdot 𝐀$	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma })}{\partial \sigma }+\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau })}{\partial \tau })+\frac{\partial A_z}{\partial z}$
Curl $\mathrm{\nabla }\times 𝐀$	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})& \widehat{𝐱}+\\ +(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})& \widehat{𝐲}+\\ +(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})& \widehat{𝐳}\end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})& \hat{\mathit{\rho }}\\ +(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })& \hat{\mathit{\varphi }}\\ +\frac{1}{\rho }(\frac{\partial \left(\rho A_{\varphi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\varphi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \varphi })& \widehat{𝒓}\\ +\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial }{\partial r}\left(r{A}_{\varphi }\right))& \hat{\mathit{\theta }}\\ +\frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })& \hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \tau }-\frac{\partial A_{\tau }}{\partial z})& \hat{\mathit{\sigma }}\\ -(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \sigma }-\frac{\partial A_{\sigma }}{\partial z})& \hat{\mathit{\tau }}\\ +\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}(\frac{\partial \left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma }\right)}{\partial \tau }-\frac{\partial \left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau }\right)}{\partial \sigma })& \widehat{𝐳}\end{array}$
Laplace operator $\mathrm{\Delta }f\equiv {\mathrm{\nabla }}^{2}f$	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\varphi }^2}$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{{\partial }^{2}f}{\partial {\sigma }^2}+\frac{{\partial }^{2}f}{\partial {\tau }^2})+\frac{{\partial }^{2}f}{\partial z^2}$
Vector Laplacian $\mathrm{\Delta }𝐀\equiv {\mathrm{\nabla }}^2𝐀$	$\mathrm{\Delta }{A}_x\widehat{𝐱}+\mathrm{\Delta }{A}_y\widehat{𝐲}+\mathrm{\Delta }{A}_z\widehat{𝐳}$	Template:Hidden begin $\begin{array}{cc}(\mathrm{\Delta }{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\varphi }}{\partial \varphi })& \hat{\mathit{\rho }}\\ +(\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \varphi })& \hat{\mathit{\varphi }}\\ +\mathrm{\Delta }{A}_z& \widehat{𝐳}\end{array}$ Template:Hidden end	Template:Hidden begin $\begin{array}{cc}(\mathrm{\Delta }{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi })& \widehat{𝒓}\\ +(\mathrm{\Delta }{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\varphi }}{\partial \varphi })& \hat{\mathit{\theta }}\\ +(\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{r^2{\sin }^2\theta }+\frac{2}{r^2\sin \theta }\frac{\partial A_r}{\partial \varphi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \varphi })& \hat{\mathit{\varphi }}\end{array}$ Template:Hidden end
Material derivative $(𝐀\cdot \mathrm{\nabla })𝐁$	Template:Hidden begin $\begin{array}{cc}(A_x\frac{\partial B_x}{\partial x}+A_y\frac{\partial B_x}{\partial y}+A_z\frac{\partial B_x}{\partial z})& \widehat{𝐱}\\ +(A_x\frac{\partial B_y}{\partial x}+A_y\frac{\partial B_y}{\partial y}+A_z\frac{\partial B_y}{\partial z})& \widehat{𝐲}\\ +(A_x\frac{\partial B_z}{\partial x}+A_y\frac{\partial B_z}{\partial y}+A_z\frac{\partial B_z}{\partial z})& \widehat{𝐳}\end{array}$ Template:Hidden end	Template:Hidden begin $\begin{array}{cc}(A_{\rho }\frac{\partial B_{\rho }}{\partial \rho }+\frac{A_{\varphi }}{\rho }\frac{\partial B_{\rho }}{\partial \varphi }+A_z\frac{\partial B_{\rho }}{\partial z}-\frac{A_{\varphi }{B}_{\varphi }}{\rho })& \hat{\mathit{\rho }}\\ +(A_{\rho }\frac{\partial B_{\varphi }}{\partial \rho }+\frac{A_{\varphi }}{\rho }\frac{\partial B_{\varphi }}{\partial \varphi }+A_z\frac{\partial B_{\varphi }}{\partial z}+\frac{A_{\varphi }{B}_{\rho }}{\rho })& \hat{\mathit{\varphi }}\\ +(A_{\rho }\frac{\partial B_z}{\partial \rho }+\frac{A_{\varphi }}{\rho }\frac{\partial B_z}{\partial \varphi }+A_z\frac{\partial B_z}{\partial z})& \widehat{𝐳}\end{array}$ Template:Hidden end	Template:Hidden begin $\begin{array}{cc}(A_r\frac{\partial B_r}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_r}{\partial \theta }+\frac{A_{\varphi }}{r\sin \theta }\frac{\partial B_r}{\partial \varphi }-\frac{A_{\theta }{B}_{\theta }+A_{\varphi }{B}_{\varphi }}{r})& \widehat{𝒓}\\ +(A_r\frac{\partial B_{\theta }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\theta }}{\partial \theta }+\frac{A_{\varphi }}{r\sin \theta }\frac{\partial B_{\theta }}{\partial \varphi }+\frac{A_{\theta }{B}_r}{r}-\frac{A_{\varphi }{B}_{\varphi }\cot \theta }{r})& \hat{\mathit{\theta }}\\ +(A_r\frac{\partial B_{\varphi }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\varphi }}{\partial \theta }+\frac{A_{\varphi }}{r\sin \theta }\frac{\partial B_{\varphi }}{\partial \varphi }+\frac{A_{\varphi }{B}_r}{r}+\frac{A_{\varphi }{B}_{\theta }\cot \theta }{r})& \hat{\mathit{\varphi }}\end{array}$ Template:Hidden end
Differential displacement	$d𝐥=dx\widehat{𝐱}+dy\widehat{𝐲}+dz\widehat{𝐳}$	$d𝐥=d\rho \hat{\mathit{\rho }}+\rho d\varphi \hat{\mathit{\varphi }}+dz\widehat{𝐳}$	$d𝐥=dr\widehat{𝐫}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\varphi \hat{\mathit{\varphi }}$	$d𝐥=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \hat{\mathit{\tau }}+dz\widehat{𝐳}$
Differential normal area $d𝐒$	$\begin{array}{cc}dydz& \widehat{𝐱}\\ +dxdz& \widehat{𝐲}\\ +dxdy& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\rho d\varphi dz& \hat{\mathit{\rho }}\\ +d\rho dz& \hat{\mathit{\varphi }}\\ +\rho d\rho d\varphi & \widehat{𝐳}\end{array}$	$\begin{array}{cc}{r}^2\sin \theta d\theta d\varphi & \widehat{𝐫}\\ +r\sin \theta drd\varphi & \hat{\mathit{\theta }}\\ +rdrd\theta & \hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}\sqrt{{\sigma }^2+{\tau }^2}d\tau dz& \hat{\mathit{\sigma }}\\ +\sqrt{{\sigma }^2+{\tau }^2}d\sigma dz& \hat{\mathit{\tau }}\\ +({\sigma }^2+{\tau }^2)d\sigma d\tau & \widehat{𝐳}\end{array}$
Differential volume $dV$	$dxdydz$	$\rho d\rho d\varphi dz$	$r^2\sin \theta drd\theta d\varphi$	$({\sigma }^2+{\tau }^2)d\sigma d\tau dz$
Non-trivial calculation rules: $\mathrm{div}\mathrm{grad}f\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }f={\mathrm{\nabla }}^{2}f\equiv \mathrm{\Delta }f$ $\mathrm{curl}\mathrm{grad}f\equiv \mathrm{\nabla }\times \mathrm{\nabla }f=\mathrm{𝟎}$ $\mathrm{div}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$ $\mathrm{curl}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$ (Lagrange's formula for del) $\mathrm{\Delta }(fg)=f\mathrm{\Delta }g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g\mathrm{\Delta }f$

Original Wikipedia Table

• from https://en.wikipedia.org/w/index.php?title=Del_in_cylindrical_and_spherical_coordinates&oldid=612099218
• Please do not edit or modify this table without very good reason

Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates

Operation	Cartesian coordinates (x, y, z)	Cylindrical coordinates (ρ, ϕ, z)	Spherical coordinates (r, θ, ϕ)	Parabolic cylindrical coordinates (σ, τ, z)
Definition of coordinates	$\begin{array}{cc}\rho & =\sqrt{x^2+y^2}\\ \varphi & =\arctan (y/x)\\ z& =z\end{array}$	$\begin{array}{cc}x& =\rho \cos \varphi \\ y& =\rho \sin \varphi \\ z& =z\end{array}$	$\begin{array}{cc}x& =r\sin \theta \cos \varphi \\ y& =r\sin \theta \sin \varphi \\ z& =r\cos \theta \end{array}$	$\begin{array}{cc}x& =\sigma \tau \\ y& =\frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =z\end{array}$
	$\begin{array}{cc}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arccos (z/r)\\ \varphi & =\arctan (y/x)\end{array}$	$\begin{array}{cc}r& =\sqrt{{\rho }^2+z^2}\\ \theta & =\arctan (\rho /z)\\ \varphi & =\varphi \end{array}$	$\begin{array}{cc}\rho & =r\sin \theta \\ \varphi & =\varphi \\ z& =r\cos \theta \end{array}$	$\begin{array}{cc}\rho \cos \varphi & =\sigma \tau \\ \rho \sin \varphi & =\frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =z\end{array}$
Definition of unit vectors	$\begin{array}{cc}\hat{\mathit{\rho }}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \hat{\mathit{\varphi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\cos \varphi \hat{\mathit{\rho }}-\sin \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐲}& =\sin \varphi \hat{\mathit{\rho }}+\cos \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\sin \theta \cos \varphi \widehat{𝒓}+\cos \theta \cos \varphi \hat{\mathit{\theta }}-\sin \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐲}& =\sin \theta \sin \varphi \widehat{𝒓}+\cos \theta \sin \varphi \hat{\mathit{\theta }}+\cos \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝒓}-\sin \theta \hat{\mathit{\theta }}\end{array}$	$\begin{array}{cc}\hat{\mathit{\sigma }}& =\frac{\tau \widehat{𝐱}-\sigma \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \hat{\mathit{\tau }}& =\frac{\sigma \widehat{𝐱}+\tau \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$
	$\begin{array}{cc}\widehat{𝐫}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}+z\widehat{𝐳}}{\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{xz\widehat{𝐱}+yz\widehat{𝐲}-(x^2+y^2)\widehat{𝐳}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\varphi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\end{array}$	$\begin{array}{cc}\widehat{𝐫}& =\frac{\rho \hat{\mathit{\rho }}+z\widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{z\hat{\mathit{\rho }}-\rho \widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}\hat{\mathit{\rho }}& =\sin \theta \widehat{𝐫}+\cos \theta \hat{\mathit{\theta }}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝐫}-\sin \theta \hat{\mathit{\theta }}\end{array}$	$\begin{array}{}\end{array}$
A vector field $𝐀$	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\widehat{𝐳}$	$A_r\widehat{𝒓}+A_{\theta }\hat{\mathit{\theta }}+A_{\varphi }\hat{\mathit{\varphi }}$	$A_{\sigma }\hat{\mathit{\sigma }}+A_{\tau }\hat{\mathit{\tau }}+A_{\varphi }\widehat{𝐳}$
Gradient $\mathrm{\nabla }f$	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial r}\widehat{𝒓}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}$	$\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \sigma }\hat{\mathit{\sigma }}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \tau }\hat{\mathit{\tau }}+\frac{\partial f}{\partial z}\widehat{𝐳}$
Divergence $\mathrm{\nabla }\cdot 𝐀$	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma })}{\partial \sigma }+\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau })}{\partial \tau })+\frac{\partial A_z}{\partial z}$
Curl $\mathrm{\nabla }\times 𝐀$	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})& \widehat{𝐱}+\\ +(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})& \widehat{𝐲}+\\ +(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})& \widehat{𝐳}\end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})& \hat{\mathit{\rho }}\\ +(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })& \hat{\mathit{\varphi }}\\ +\frac{1}{\rho }(\frac{\partial \left(\rho A_{\varphi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\varphi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \varphi })& \widehat{𝒓}\\ +\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial }{\partial r}\left(r{A}_{\varphi }\right))& \hat{\mathit{\theta }}\\ +\frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })& \hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \tau }-\frac{\partial A_{\tau }}{\partial z})& \hat{\mathit{\sigma }}\\ -(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \sigma }-\frac{\partial A_{\sigma }}{\partial z})& \hat{\mathit{\tau }}\\ +\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}(\frac{\partial \left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma }\right)}{\partial \tau }-\frac{\partial \left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau }\right)}{\partial \sigma })& \widehat{𝐳}\end{array}$
Laplace operator $\mathrm{\Delta }f\equiv {\mathrm{\nabla }}^{2}f$	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\varphi }^2}$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{{\partial }^{2}f}{\partial {\sigma }^2}+\frac{{\partial }^{2}f}{\partial {\tau }^2})+\frac{{\partial }^{2}f}{\partial z^2}$
Vector Laplacian $\mathrm{\Delta }𝐀\equiv {\mathrm{\nabla }}^2𝐀$	$\mathrm{\Delta }{A}_x\widehat{𝐱}+\mathrm{\Delta }{A}_y\widehat{𝐲}+\mathrm{\Delta }{A}_z\widehat{𝐳}$	Template:Collapsible section	Template:Collapsible section
Material derivative $(𝐀\cdot \mathrm{\nabla })𝐁$	Template:Collapsible section	Template:Collapsible section	Template:Collapsible section
Differential displacement	$d𝐥=dx\widehat{𝐱}+dy\widehat{𝐲}+dz\widehat{𝐳}$	$d𝐥=d\rho \hat{\mathit{\rho }}+\rho d\varphi \hat{\mathit{\varphi }}+dz\widehat{𝐳}$	$d𝐥=dr\widehat{𝐫}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\varphi \hat{\mathit{\varphi }}$	$d𝐥=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \hat{\mathit{\tau }}+dz\widehat{𝐳}$
Differential normal area $d𝐒$	$\begin{array}{cc}dydz& \widehat{𝐱}\\ +dxdz& \widehat{𝐲}\\ +dxdy& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\rho d\varphi dz& \hat{\mathit{\rho }}\\ +d\rho dz& \hat{\mathit{\varphi }}\\ +\rho d\rho d\varphi & \widehat{𝐳}\end{array}$	$\begin{array}{cc}{r}^2\sin \theta d\theta d\varphi & \widehat{𝐫}\\ +r\sin \theta drd\varphi & \hat{\mathit{\theta }}\\ +rdrd\theta & \hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}\sqrt{{\sigma }^2+{\tau }^2}d\tau dz& \hat{\mathit{\sigma }}\\ +\sqrt{{\sigma }^2+{\tau }^2}d\sigma dz& \hat{\mathit{\tau }}\\ +({\sigma }^2+{\tau }^2)d\sigma d\tau & \widehat{𝐳}\end{array}$
Differential volume $dV$	$dxdydz$	$\rho d\rho d\varphi dz$	$r^2\sin \theta drd\theta d\varphi$	$({\sigma }^2+{\tau }^2)d\sigma d\tau dz$
Non-trivial calculation rules: $\mathrm{div}\mathrm{grad}f\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }f={\mathrm{\nabla }}^{2}f\equiv \mathrm{\Delta }f$ $\mathrm{curl}\mathrm{grad}f\equiv \mathrm{\nabla }\times \mathrm{\nabla }f=\mathrm{𝟎}$ $\mathrm{div}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$ $\mathrm{curl}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$ (Lagrange's formula for del) $\mathrm{\Delta }(fg)=f\mathrm{\Delta }g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g\mathrm{\Delta }f$